Rate of change

The essentials

The derivative dy/dx tells how fast y changes compared with x. If a quantity y depends on x, then the value of dy/dx at x = x₀ is the rate of change of y with respect to x at that point. When both quantities change with time t, the chain rule links their rates: dy/dt = (dy/dx)(dx/dt). This lets you find how fast the area of a spreading circle or the volume of an inflating balloon grows when you know how fast the radius changes.

Key points

• Circle: A = πr², so dA/dr = 2πr and dA/dt = 2πr (dr/dt).
• Sphere: V = (4/3)πr³, so dV/dt = 4πr² (dr/dt); surface area S = 4πr², so dS/dt = 8πr (dr/dt).
• Cube of side x: V = x³, so dV/dt = 3x² (dx/dt).
• Motion: if s(t) is position, v = ds/dt and acceleration = dv/dt.
• Units of a rate are (units of y) per (unit of x), for example cm²/s.

Exam tips

• Write down the formula connecting the quantities, differentiate with respect to t, then substitute the given values. Substitute only after differentiating.
• Read carefully whether the question wants a rate with respect to time or with respect to another variable such as the radius.

Common mistakes

• Substituting the particular value (like r = 5) before differentiating, which turns a variable into a constant.
• Dropping the units or giving cm³ for a rate that should be cm³/s.